* Dual category and dual object @ 2010-09-04 10:46 David Leduc 2010-09-04 14:46 ` Michael Barr ` (4 more replies) 0 siblings, 5 replies; 8+ messages in thread From: David Leduc @ 2010-09-04 10:46 UTC (permalink / raw) To: categories Are the notions of dual category and dual object related? If not, are there any good reasons to use the word "dual" for both notions? [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Dual category and dual object 2010-09-04 10:46 Dual category and dual object David Leduc @ 2010-09-04 14:46 ` Michael Barr 2010-09-04 16:38 ` Toby Bartels ` (3 subsequent siblings) 4 siblings, 0 replies; 8+ messages in thread From: Michael Barr @ 2010-09-04 14:46 UTC (permalink / raw) To: David Leduc; +Cc: categories On Sat, 4 Sep 2010, David Leduc wrote: > Are the notions of dual category and dual object related? > If not, are there any good reasons to use the word "dual" for both notions? > > Of course they're related. Under a categorical duality, the object in one category that corresponds to it in the other is the dual object. E.g. under the duality between sets and complete atomic boolean algebras, the object dual to to the set S is 2^S and the object dual to a CABA B is the set of atoms of B. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Dual category and dual object 2010-09-04 10:46 Dual category and dual object David Leduc 2010-09-04 14:46 ` Michael Barr @ 2010-09-04 16:38 ` Toby Bartels 2010-09-05 0:47 ` Michael Barr [not found] ` <Pine.LNX.4.64.1009042045000.20602@msr03.math.mcgill.ca> 2010-09-04 19:41 ` Aleks Kissinger ` (2 subsequent siblings) 4 siblings, 2 replies; 8+ messages in thread From: Toby Bartels @ 2010-09-04 16:38 UTC (permalink / raw) To: categories; +Cc: David Leduc David Leduc wrote: >Are the notions of dual category and dual object related? The concept of dual vector space (which may be the historical original) is an example of both, although in slightly different ways. A vector space is an object in Vect, and its dual is a dual object; taking a vector space to its dual extends to a contravariant functor, which is the same thing as a functor on a dual category. >If not, are there any good reasons to use the word "dual" for both notions? Perhaps we should say "adjoint object" instead of "dual object". Every monoidal category can be interpreted as a 2-category, and a dual object in the monoidal category generalises to an adjoint morphism in the 2-category. On the other hand, we can say "opposite category" instead of "dual category", especially since we denote the dual category of C by C^{op}. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Dual category and dual object 2010-09-04 16:38 ` Toby Bartels @ 2010-09-05 0:47 ` Michael Barr [not found] ` <Pine.LNX.4.64.1009042045000.20602@msr03.math.mcgill.ca> 1 sibling, 0 replies; 8+ messages in thread From: Michael Barr @ 2010-09-05 0:47 UTC (permalink / raw) To: Toby Bartels; +Cc: categories, David Leduc On Sat, 4 Sep 2010, Toby Bartels wrote: > David Leduc wrote: > >> Are the notions of dual category and dual object related? >....................................................... > > On the other hand, we can say "opposite category" instead of "dual category", > especially since we denote the dual category of C by C^{op}. > So you would say that complete atomic boolean algebras is just Set^{op}? Well I wouldn't. They are, of course, equivalent, but not the same. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Dual category and dual object [not found] ` <Pine.LNX.4.64.1009042045000.20602@msr03.math.mcgill.ca> @ 2010-09-05 0:54 ` Toby Bartels 0 siblings, 0 replies; 8+ messages in thread From: Toby Bartels @ 2010-09-05 0:54 UTC (permalink / raw) To: categories; +Cc: Michael Barr Michael Barr wrote: >Toby Bartels wrote: >>David Leduc wrote: >>>Are the notions of dual category and dual object related? >>On the other hand, we can say "opposite category" instead of "dual category", >>especially since we denote the dual category of C by C^{op}. >So you would say that complete atomic boolean algebras is just Set^{op}? >Well I wouldn't. They are, of course, equivalent, but the same. I would say both "The dual category of Set is Set^{op}." and "CABA is a dual category of Set." (and also "CABA is an opposite category of Set."). Actually, I would probably do the grammar differently: "CABA is a category dual [or "opposite"] to Set.". Leduc's original question didn't have an article in it; you seem to have interpreted is with "a" while I interpreted it with "the". (We also interpreted "dual object" differently.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Dual category and dual object 2010-09-04 10:46 Dual category and dual object David Leduc 2010-09-04 14:46 ` Michael Barr 2010-09-04 16:38 ` Toby Bartels @ 2010-09-04 19:41 ` Aleks Kissinger [not found] ` <3E056346-48EE-4D5C-A2FD-8008A722583A@mq.edu.au> [not found] ` <AANLkTikBACfX_x3DX2pUL6zSGc_5xYoz42q3Gbf3Hpby@mail.gmail.com> 4 siblings, 0 replies; 8+ messages in thread From: Aleks Kissinger @ 2010-09-04 19:41 UTC (permalink / raw) To: David Leduc; +Cc: categories In the (bi)category Prof of categories and profunctors, the dual of an object is the dual category. Profunctors most certainly came later than the notions of categorical dual and dual objects (or at least their concrete counterparts, dual spaces), so this might just be a happy coincidence. Aleks On Sat, Sep 4, 2010 at 5:46 AM, David Leduc <david.leduc6@googlemail.com> wrote: > Are the notions of dual category and dual object related? > If not, are there any good reasons to use the word "dual" for both notions? > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Dual category and dual object [not found] ` <3E056346-48EE-4D5C-A2FD-8008A722583A@mq.edu.au> @ 2010-09-05 1:50 ` David Leduc 0 siblings, 0 replies; 8+ messages in thread From: David Leduc @ 2010-09-05 1:50 UTC (permalink / raw) To: Ross Street, Aleks Kissinger; +Cc: categories Could you please spell out what is the unit in Prof (It is not Set, is it?) and what are the units and counits for dual objects? Thanks for your help. On Sun, Sep 5, 2010, Ross Street <ross.street@mq.edu.au> wrote: > On 05/09/2010, at 5:41 AM, Aleks Kissinger wrote: >> In the (bi)category Prof of categories and profunctors, the dual of an >> object is the dual category. Profunctors most certainly came later >> than the notions of categorical dual and dual objects (or at least >> their concrete counterparts, dual spaces), so this might just be a >> happy coincidence. > > Very well put! > I might add that an extra point needed is that Prof is a monoidal bicategory > where the tensor product is the cartesian product of categories (it is not > the cartesian product in Prof). And yes, Prof is compact, autonomous, > rigid, whichever word you prefer, and the dual in Prof of a category A > is A^{op}. In reading the literature, note that other names for Prof are > Dist, Bimod and Mod. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Dual category and dual object [not found] ` <AANLkTikBACfX_x3DX2pUL6zSGc_5xYoz42q3Gbf3Hpby@mail.gmail.com> @ 2010-09-05 2:30 ` Aleks Kissinger 0 siblings, 0 replies; 8+ messages in thread From: Aleks Kissinger @ 2010-09-05 2:30 UTC (permalink / raw) To: David Leduc; +Cc: Ross Street, categories I originally thought it was Set, but this doesn't seem to make sense using the cartesian product of categories as the monoidal product in Prof. If this were the case, candidates for the unit and counit would be the upper and lower stars of the hom functor. However, let the 1-object category be the monoidal product. Then, since 1 X C^op X C is isomorphic to C^op X C, we can regard the hom functor of C as a profunctor 1 --|--> C^op X C, Hom_C : 1 X C^op X C --> Set and similarly, the hom functor of C^op as a profunctor C X C^op --|--> 1. I saw this briefly mentioned here: http://ncatlab.org/nlab/show/trace+of+a+category Does anyone know if there is a more fleshed out treatment of this somewhere? Also, do the "lifted" versions of the hom functor (upper and lower star) serve some structural purpose in Prof? Aleks On Sat, Sep 4, 2010 at 8:50 PM, David Leduc <david.leduc6@googlemail.com> wrote: > Could you please spell out what is the unit in Prof (It is not Set, is > it?) and what are the units and counits for dual objects? > > Thanks for your help. > > > On Sun, Sep 5, 2010, Ross Street <ross.street@mq.edu.au> wrote: >> On 05/09/2010, at 5:41 AM, Aleks Kissinger wrote: >>> In the (bi)category Prof of categories and profunctors, the dual of an >>> object is the dual category. Profunctors most certainly came later >>> than the notions of categorical dual and dual objects (or at least >>> their concrete counterparts, dual spaces), so this might just be a >>> happy coincidence. >> >> Very well put! >> I might add that an extra point needed is that Prof is a monoidal bicategory >> where the tensor product is the cartesian product of categories (it is not >> the cartesian product in Prof). And yes, Prof is compact, autonomous, >> rigid, whichever word you prefer, and the dual in Prof of a category A >> is A^{op}. In reading the literature, note that other names for Prof are >> Dist, Bimod and Mod. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
end of thread, other threads:[~2010-09-05 2:30 UTC | newest] Thread overview: 8+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2010-09-04 10:46 Dual category and dual object David Leduc 2010-09-04 14:46 ` Michael Barr 2010-09-04 16:38 ` Toby Bartels 2010-09-05 0:47 ` Michael Barr [not found] ` <Pine.LNX.4.64.1009042045000.20602@msr03.math.mcgill.ca> 2010-09-05 0:54 ` Toby Bartels 2010-09-04 19:41 ` Aleks Kissinger [not found] ` <3E056346-48EE-4D5C-A2FD-8008A722583A@mq.edu.au> 2010-09-05 1:50 ` David Leduc [not found] ` <AANLkTikBACfX_x3DX2pUL6zSGc_5xYoz42q3Gbf3Hpby@mail.gmail.com> 2010-09-05 2:30 ` Aleks Kissinger
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